Gauging the Performance of Density Functionals for Lanthanide


Feb 5, 2016 - ... hybrid functionals perform worse than functionals without exact exchange, and TPSS performs the best overall for the Ln54 set with a...
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Gauging the performance of density functionals for lanthanide-containing molecules Stephanie Grimmel, George Schoendorff, and Angela K Wilson J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.5b01193 • Publication Date (Web): 05 Feb 2016 Downloaded from http://pubs.acs.org on February 8, 2016

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Gauging the performance of density functionals for lanthanide-containing molecules Stephanie Grimmel1,2, George Schoendorff1,3, and Angela K. Wilson1,3 * 1 2 3

Department of Chemistry and Center for Advanced Scientific Computing and Modeling (CASCaM), University of North Texas, Denton, Texas 76203-5017, United States Department of Chemistry and Biochemistry, Karlsruher Institut für Technologie, Kaiserstraße 12, 76131 Karlsruhe, Germany Department of Chemistry, Michigan State University, East Lansing, Michigan 488241322, United States

Abstract Several density functional approaches have been considered for their ability to predict enthalpies of formation and bond dissociation energies for lanthanide-containing molecules. To enable comparison with experiment, the Ln54 set, introduced here, is compiled to include lanthanides both in the common 3+ oxidation state as well as in more exotic oxidation states. Due to the magnitude of the experimental uncertainties a “lanthanide chemical accuracy” of 5.0 kcal mol-1 is proposed. The density functionals considered span the full range of complexity from LDA through double hybrids. The performance of the density functionals is assessed for each class of lanthanide-containing molecules and for the Ln54 molecule set overall. In general, hybrid functionals perform worse than functionals without exact exchange, and TPSS performs the best overall for the Ln54 set with a MAD of 19.2 kcal mol-1 and MSD of -1.9 kcal mol-1.

Keywords Lanthanide, lanthanoid, Ln54, density functional theory

*

Author to whom correspondence should be addressed: [email protected]

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Introduction The computation of properties of lanthanide-containing molecules pose numerous challenges.

As for all heavy elements, the large number of core electrons can result in

substantial computational cost in terms of computer memory and CPU time. Often the large number of core electrons, and the basis functions needed to describe the core, can be mitigated via the use of a pseudopotential such as an effective core potential (ECP). 1 However, due to the compact nature of the 4f orbitals, the subvalence 5s and 5p should be treated explicitly and included in the correlation space when using ab initio methods such as coupled cluster methods or perturbation theory. Moreover, additional electrons deeper in the core may need to be correlated in order to obtain the desired accuracy for the properties computed. 2-4 Additionally, computation of lanthanides and lanthanide-containing molecules must include a treatment of relativistic effects either explicitly via methods such as Douglas-Kroll, 5-7 zeroth-order relativistic approximation (ZORA), 8, 9 infinite order two component, 10, 11 etc., or implicitly through the use of a relativistic ECP core. 1 Lanthanides often require a treatment of static correlation effects via multireference methods. This is due primarily to the partial occupation of the 4f shell that dominates the multireference character of lanthanides in the common 3+ oxidation state. Yet the pseudo corelike nature of the 4f orbitals results primarily in ionic bonding with excited state surfaces that tend to be parallel to the ground state. 12, 13 In addition to the 4f degeneracy, lanthanides in lowvalency states also have near degeneracies between the 6s and 5d shells. Occupation of the 6s and 5d shells tends to result in covalent bonding, and the difference in the radial extent of the 6s and 5d orbitals leads to excited states surfaces that are not parallel to the ground state and hence numerous surface crossings occur. 2 The large number of low-lying excited states is a hallmark

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of lanthanide chemistry and often leads to complications when single reference wavefunction methods are employed since it is possible to converge to one of the nearby excited states instead of the ground state. For example, the cerium atom is problematic for single reference methods because the ground state of the neutral cerium atom is an open-shell singlet state, 1G, with an electronic configuration of 4f15d16s2. Low spin open-shell species are problematic since most implementations of open-shell DFT require high spin. Therefore, the analogous high spin triplet state with 4f15d16s2 occupation must be considered instead of the 1G ground state. However, there are three low-lying triplet states with this electronic configuration that differ only in the f orbital occupation, i.e. 3F, 3H, and 3G, and these states range in energy from 229 cm-1 to 4199 cm-1 above the 1G ground state. 14 Thus, the use of a triplet state when computing the energy neutral cerium atom will introduce error into computed energetic properties (e.g. enthapies of formation and bond dissociation energies) that is beyond the inherent error associated with the method employed. A further complication arises from the fact that there is a quintet state, 5H, within this energy range. This state contributes to spin contamination when an unrestricted implementation is used. In fact the percent spin contamination (Eq. 1) can be used as a guide to determine if a species such as the cerium atom has low-lying excited states of higher spin.

% Spin Contamination =

( Sˆ

calculated



− Sˆ

exact

) × 100

(1)

exact

Given the scaling of multireference wavefunction methods versus single reference € methods, single reference methods are often employed for the practical calculations of

lanthanide-containing molecules despite the challenges associated with modeling f-block compounds. Single reference density functional theory (DFT) is by far the most popular method used for modeling molecules with f-block elements. 15, 16 Depending on the oxidation state of the

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lanthanide, molecules containing lanthanides near the beginning, middle, or end of the series tend to be the most amenable to single reference methods due to the empty, half-filled, or full 4f shell. Lanthanides in the middle of the first or second half of the series often exhibit the most significant multireference character due to partial occupation of the 4f shell, thus, it is particularly important to assess the efficacy of single reference methods for their ability to provide reliable computed properties before delving into practical application., In this work, an assessment of the performance of a broad range of density functionals for computed energetic properties of lanthanides has been done. To accomplish this, a set of 54 lanthanide-containing molecules, the Ln54 molecule set, is introduced wherein the enthalpies of formation or bond dissociation energies are known experimentally.

Computational Methods All calculations were performed using the NWChem computational chemistry software package. 17 The Sapporo-DKH3-TZP-2012 basis sets were used for all lanthanides, 18 the ccpVTZ basis set was used for 2p atoms, 19 and the cc-pV(T+d)Z-DK basis set was used for 3p atoms. 20-24 The third order Douglas-Kroll (DK3) method was used for an a priori treatment of

v v relativistic effects. 5-7 The implementation of DK3† includes the p × Vp cross product terms in the Hamiltonian, thereby implicitly accounting for vector relativistic effects. The results remain

€ J-averaged, though, due to the neglect of the odd terms. 25-27 All geometry optimizations were performed using the PBE0 functional, 28 and the thermal corrections to the total energies were obtained at the PBE0 level.

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The geometries of the

lanthanide trihalides molecules were constrained to D3h symmetry since the barrier to inversion



DK3full in NWChem

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is small with most measured and computed barriers less than 100 cm-1.

30-32

Enthalpies of

formation and bond dissociation energies were computed for molecules in the Ln54 set using a variety of density functionals spanning a full range of functional complexity from the local spin density approximation through double hybrids. The LDA functional employed was the SVWN functional. 33, 34 GGA functionals used were BP86 35, 36, BLYP 35, 37, PW91 38, and PBE 39. The meta-GGA functionals employed were TPSS

40

and M06-L.

41

Hybrid-GGA functionals used

were PBE0, B3LYP 33, 35, 37, 42, 43, BHLYP 37, 44, B3P86 35, 36, B97-1 45, 46, MPW1K 47, and X3LYP 34, 35, 37, 38

. The hybrid meta-GGA functionals used were TPSSH 48, M06 49, and M06-2X. 49 The

range separated hybrid functionals used were SSB-D 50, 51, B97-D, CAM-B3LYP 52, and M11 53 while B2PLYP 35, 37, 54, 55 was the double hybrid functional employed. The performance of each functional was assessed for its ability to reproduce enthalpies of formation from experiment and bond dissociation energies.

The sublimation enthalpies of the lanthanides used in the

computation of the enthalpies of formation are available in the Supplementary Material (Table S4). 56-58 Restricted open-shell DFT was used for calculations with all functionals except for B2PLYP.

Unrestricted open-shell DFT was used for B2PLYP since the MP2 module in

NWChem does not support a restricted reference.

The electronic states that have been

determined in this study correspond to the electronic states of the atoms and molecules that have been determined experimentally. In situations where the ground electronic state is not known, an ionic model was assumed wherein the ligands are closed shell anions and the lanthanide cation has all the spin density with the electrons occupying the same orbitals as the lanthanide ion. For example, LnO compounds were computed as if they were Ln2+O2- with the unpaired electrons occupying the same orbitals as they would with the bare Ln2+ ion. 59, 60

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Results and Discussion In order to gauge the utility of any computational method, a suitable set of molecules is needed. To this end, the Ln54 molecule set is introduced which has been compiled based upon 54 lanthanide-containing molecules for which the enthalpy of formation, ∆Hf(298), or bond dissociation energy, D0, is known from experiment. 61-66 To further gauge performance based on the molecular environment, the Ln54 molecule set is divided into three subsets (Tables S1, S2 and S3 in the Supplementary Material). The Ln54HFa molecule set consists of 15 molecules with known enthalpies of formation where the lanthanide is formally in the 3+ oxidation state. The maximum uncertainty from experiment is 3.2 kcal mol-1 for NdF3 and TbF3 and the average uncertainty from experiment is 2.0 kcal mol-1. The Ln54HFb set of molecules is composed of 31 molecules with the lanthanide in a formal oxidation state other than 3+ for which the enthalpies of formation from experiment are known. The uncertainty from experiment is greater for the Ln54HFb set than it is for the Ln54HFa set with a maximum uncertainty of 15 kcal mol-1 for LaF2 and an average uncertainty of 4.9 kcal mol-1. The final subset of Ln54 is the Ln54D0 set of molecules. The Ln54D0 set contains 25 diatomic molecules for which the bond dissociation energy, D0, is known from experiment. As for the molecules in the Ln54HFb set, the lanthanides in the Ln54D0 set formally are in a low-valency state, i.e. 1+ or 2+. The uncertainties in the values from experiment for the Ln54D0 set are on par with the Ln54HFb set with a maximum uncertainty of 11 kcal mol-1 for PrF and an average experimental uncertainty of 4.7 kcal mol-1. Taken together, the average experimental uncertainty of the entire Ln54 set of molecules is 4.5 kcal mol-1. Due to the large uncertainties of the experimental data in the Ln54HFb and Ln54D0

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sets, a new definition of chemical accuracy (“lanthanide chemical accuracy”) of 5 kcal mol-1 for energetic properties of lanthanide-containing molecules is proposed. The Ln54HFa subset of molecules The Ln54HFa subset of molecules consists of molecules with the lanthanide formally in the common 3+ oxidation state. The performance of the density functionals in the prediction of enthalpies of formation, ∆Hf(298), for the Ln54HFa subset of molecules is shown in Figure 1. The mean signed deviations (MSD) are roughly equal in magnitude to the mean absolute deviations (MAD) indicating systematic error for all density functionals with the predicted enthalpies of formation consistently lower that the experimental values. The best results were obtained with the TPSS functional, which shows a MAD of 24.9 kcal mol-1, although the SWVN result is a close second with a MAD of 25.0 kcal mol-1. Simple functionals that neglect exact exchange, i.e. LDA, GGA, and meta-GGA functionals, perform the best with errors ranging from 24.9 kcal mol-1 with TPSS to 37.8 kcal mol-1 with the SSB-D functional. The hybrid and double hybrid functionals generally resulted in the poorest performance with MADs ranging from 33.1 kcal mol-1 for M06 to 53.1 kcal mol-1 for B2PLYP with a total of six functionals producing errors in excess of 40 kcal mol-1, i.e. BHLYP, X3LYP, B3LYP, CAM-B3LYP, M06-2X, and B2PLYP. The Ln54HFb subset of molecules The Ln54HFb subset consists of molecules in which the lanthanide ion has a formal oxidation state other than the common 3+.

The performance of the density functional

approximations for the prediction of enthalpies of formation, ∆Hf(298), for the Ln54HFb subset of molecules is shown in Figure 2. In contrast with the results for the Ln54HFa subset, there is less of a systematic error in the computed enthalpies of formation. This is demonstrated by the large differences between the magnitudes of the MSDs and MADs for each functional. While

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the enthalpies of formation for the Ln54HFa subset were consistently lower than the results from experiment, there is no analogous trend with the Ln54HFb set resulting in small MSDs relative to the MADs. The best performance for the computation of the enthalpies of formation is achieved for the BP86 functional resulting in a MAD of 14.9 kcal mol-1, although PW91 and TPSS perform nearly as well with MADs of 15.1 kcal mol-1 and 16.4 kcal mol-1, respectively. The poorest performance occurs for the M06-2X functional with a MAD of 31.8 kcal mol-1, and BHLYP performs nearly as poorly with a MAD of 29.9 kcal mol-1. As for the Ln54HFa subset, the functionals without exact exchange perform the best with MADs ranging from 14.9 kcal mol1

(BP86) to 18.2 kcal mol-1 (M06-L), whereas the range for the hybrid and double hybrid

functionals is 18.5 kcal mol-1 (TPSSH) to 31.8 kcal mol-1 (M06-2X). The lanthanide monoxides, LnO, pose challenges when DFT is employed.

The

enthalpies of formation from experiment are small ranging from -0.8 kcal mol-1 for LuO to -34.8 kcal mol-1 for PrO. In general, the magnitude of the enthalpies of formation decreases across the series with MADs for the LnO molecules ranging from 19.4 kcal mol-1 (PW91 and BP86) to 56.3 kcal mol-1 (M06-2X) and MSDs that are all negative. As the magnitude of the error in the computed results can be greater than the magnitude of the enthalpy of formation, it is possible for the computed enthalpies of formation to have the wrong sign. This actually occurs for every functional tested with TbO, DyO, HoO, and ErO consistently predicted to have positive enthalpies of formation. Inclusion of exact exchange exacerbates the problem causing other LnO species to have computed positive enthalpies of formation manifested primarily for the latter half of the lanthanide series. The performance of density functionals for the prediction of enthalpies of formation for the complete Ln54HF set encompassing both the Ln54HFa and Ln54HFb sets is shown in Figure

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3. The general trend for the combined set is that functionals without exact exchange perform the best. The TPSS performed best overall with a MAD of 19.1 kcal mol-1, and the performance of PW91, SVWN, and PBE are all within 1 kcal mol-1 of the TPSS results. The Ln54D0 subset of molecules The Ln54D0 set of molecules consists of diatomic molecules with experimentally known bond dissociation energies. The performance of the density functional approximations for the computation of bond dissociation energies is shown in Figure 4. With the exception of SVWN, the MSD is always positive indicating a general tendency to overestimate the strength of the bond. The magnitude of the MSD is substantially smaller than the MAD for functionals that do not include exact exchange. This indicates that while there is a tendency to over-bind, it is not a systematic error for these functionals. In contrast, the magnitude of the MSD is almost as large as the MAD for hybrid and double hybrid functionals indicating that the inclusion of exact exchange results in systematic over-binding of the molecules. All density functionals predict low bond dissociation energies for DyO, HoO, and ErO with respect to experiment in contrast with the general trend for the Ln54D0 subset of molecules.

Inclusion of exact exchange

exacerbates the situation, with predictions of negative binding energies when MPW1K, BHLYP, and M06-2X are used, i.e. the functionals with the greatest degree of exact exchange. Moreover, DyO and ErO have high degrees of spin contamination when unrestricted open-shell DFT is used for the B2PLYP functional. The percent spin contamination is 16.8% and 50.2% for DyO and ErO, respectively. This is indicative of higher spin states in the vicinity of the computed state, and thus the assumption of ionic Ln2+O2- fails for these two molecules contributing further to the error in the computed bond dissociation energy. The results from other functionals can be improved slightly by using unrestricted open-shell DFT for DyO and ErO so that mixing with

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low-lying excited states is. Likewise, the cerium atom has a percent spin contamination of 13.1% so that the use of an unrestricted formalism for cerium-containing species also shows improvement in the computed enthalpies of formation and bond dissociation energies. When an unrestricted formalism is used the MAD for the Ln54 molecules set improves by up to 1 kcal mol-1.

However, an unrestricted method is not needed for the other species in the Ln54

molecules set since the percent spin contamination is on the order of 1% for all remaining atomic and molecular species. Overall, the functionals without exact exchange perform the best resulting in MADs ranging from 17.3 kcal mol-1 with the BP86 functional to 24.9 kcal mol-1 with the PBE functional. Hybrid and double hybrid functionals have MADs that range from 24.7 kcal mol-1 with the TPSSH functional to 52.4 kcal mol-1 with the BHLYP functional, and ten functionals have MADs in excess of 30 kcal mol-1, i.e. PBE0, BHLYP, MPW1K, B97-1, X3LYP, M06, M06-2X, M11, CAM-B3LYP, and B2PLYP. The combined Ln54 molecule set The performance of the density functional approximations for the computation of enthalpies of formation and bond dissociation energies for the combined Ln54 molecule set is shown in Figure 5. The magnitudes of the MSDs for all functionals are less than 10 kcal mol-1 and some MSDs are even within 1 kcal mol-1 of experiment. However, the MSDs rely on cancellation error, notably from the predominantly negative MSDs associated with enthalpies of formation and the predominantly positive MSDs associated with the bond dissociation energies. Thus, the best measure of the performance of the functionals is the MAD.

Overall, the

functional that performed the best for the Ln54 molecule set was the TPSS meta-GGA functional

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with a MAD of 19.2 kcal mol-1. Two of the GGA functionals performed nearly as well with MADs of 19.3 kcal mol-1 and 19.7 kcal mol-1 for the PW91 and BP86 functionals, respectively. In general, the MADs are smaller for functionals without exact exchange compared with the hybrid and double hybrid functionals. The effect of increasing levels of exact exchange is shown in the comparison of the MADs for the M06 family of density functionals since the difference between the various functionals solely can be attributed to the amount of exact exchange included within each functional. The percentage of exact exchange within the M06 family of functionals ranges from 0% - 54% (Table 1). M06-L does not include exact exchange and consistently outperforms the other M06 functionals. Inclusion of 27% exact exchange in the M06 functional increases the MAD by up to 3.8 kcal mol-1 for enthalpies of formation and 9.0 kcal mol-1 for bond dissociation energies. Doubling the percentage of exact exchange to 54%, which occurs for the M06-2X functional, increases the error further with an additional error of up to 13.4 kcal mol-1 for the enthalpies of formation and an additional 12.8 kcal mol-1 error for the bond dissociation energies. Thus, the performance of the hybrid functionals diminishes with increasing percentages of exact exchange. Moreover, when exact exchange is included, the MAD increases with increasing functional complexity and non-locality, i.e. hybrid GGA < hybrid meta-GGA < range separated hybrid GGA < double hybrid GGA. This trend holds for all but the three functionals with exact exchange contributions greater than 40%, i.e. MPW1K, BHLYP, and M06-2X. It is clear that of all the methods that include non-local effects in the functional, exact exchange has the greatest impact upon the error in the computed energetic properties with long-range corrections and perturbation correlation playing a lesser role. While seemingly counterintuitive, this result is due to the fact that the bulk of the electron density is localized at the metal center rather than the ligands. In fact, it is known that LDA becomes exact

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in the limit of an infinite nuclear charge 67 since the number of associated electrons grows with increasing nuclear charge while also becoming more localized around the highly charged nucleus. Thus the local and semi-local functionals that rely on only the occupied orbitals are best suited for such highly localized densities. Finally, the best results are consistently obtained by functionals with a minimum of empirical parameters such as PW91, BP86, and TPSS. Since parameterization is typically performed for light main group elements, it is not surprising that the associated parameters may fail at the bottom of the periodic table.

Conclusions The Ln54 set of lanthanide-containing molecules has been introduced herein as a gauge of the performance of density functional approximations for the prediction of enthalpies of formation and bond dissociation energies. The Ln54 molecule set is divided into three subsets characterized by the energetic property and by the formal oxidation state of the lanthanide. The Ln54HFa subset includes molecules with lanthanides formally in the 3+ oxidation state and an average experimental uncertainty of 2.0 kcal mol-1 for the enthalpies of formation. The second subset, Ln54HFb, includes molecules lanthanides in low-valency oxidation states with an average experimental uncertainty of 4.9 kcal mol-1 for the enthalpies of formation. Finally, the Ln54D0 subset of molecules consists of lanthanide-containing diatomic molecules with an average experimental uncertainty of 4.7 kcal mol-1 for bond dissociation energies. Because of the large uncertainties associated with energetic properties from experiment of lanthanidecontaining molecules, a revised definition of chemical accuracy is proposed: a lanthanide chemical accuracy for computed energies properties of lanthanide-containing molecules is 5.0 kcal mol-1.

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The density functional approximations considered spanned the full range of functional complexity. In general the predicted enthalpies of formation were too low compared with experiment for all functionals tested. And with the exception of the SVWN functional, all functionals tend to overestimate the bond dissociation energy.

It is shown that the best

performance for computing both the enthalpies of formation and bond dissociation energies is obtained with functionals that neglect exact exchange. For functionals that do include exact exchange, there is a clear trend of increasing MADs with increasing fractions of exact exchange. This is due to the fact that for heavy elements such as the lanthanides, the bulk of the electron density is localized on the heavy element. Therefore, local functionals that depend only upon occupied orbitals perform better than non-local functionals such as hybrids, double hybrids, and range-separated hybrids. The best performance is obtained with the TPSS meta-GGA functional with MADs of 19.1 kcal mol-1 and 19.2 kcal mol-1 for the computation of enthalpies of formation and bond dissociation energies, respectively, and an overall MAD of 19.2 kcal mol-1 for the full Ln54 molecule set.

Supporting Information Computed enthalpies of formation and bond dissociation energies for each density functional as well as details of the Ln54 molecule set and the Ln54HFa, Ln54HFb, and Ln54D0 subsets are available in the supporting information. This material is available free of charge via the internet at http://pubs.acs.org.

Acknowledgements

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This project was funded by the National Science Foundation under Gran No. CHE- CHE1362479. Computing resources were provided by the Computing and Information Technology Center at the University of North Texas.

Additional support was provided by the U.S.

Department of Energy (DOE) for the Center for Advanced Scientific Computing and Modeling (CASCaM) and by the Deutscher Akademischer Austausch Dienst (DAAD) RISE program.

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15. Platas-Iglesias, C.; Roca-Sabio, A.; Regueiro-Figueroa, M.; Esteban-Gómez, D.; de Blas, A.; Rodríguez-Blas, R. Applications of density functional theory (DFT) to investigate the structural, spectroscopic and magnetic properties of lanthanide(III) complexes. Current Inorg. Chem. 2011, 1, 91-116. 16. Heiberg, H.; Gropen, O.; Laerdahl, J. K.; Swang, O.; Wahlgren, U. The performance of density functional theory for LnF (Ln = Nd, Eu, Gd, Yb) and YbH. Theor. Chem. Acc. 2003, 110, 118-125. 17. Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; van Dam, H. J. J.; Wang, D.; Mieplocha, J.; Aprá, E.; Windus, T. L., et al. NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations. Comput. Phys. Commun. 2010, 181, 1477-1489. 18. Sekiya, M.; Noro, T.; Koga, T.; Shimazaki, T. Relativistic segmented contraction basis sets with core-valence correlation effects for atoms La through Lu: Sapporo-DK-nZP sets (n = D, T, Q) Theory. Chem. Acc. 2012, 131, 1247. 19. Dunning Jr., T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. 20. Woon, D. E.; Dunning Jr., T. H. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358-1371. 21. Dunning Jr., T. H., Peterson, K. A.; Wilson, A. K. Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited. J. Chem. Phys. 2001, 114, 9244-9253. 22. de Jong, W. A.; Harrison, R. J.; Dixon, D. A. Parallel Douglas-Kroll energy and gradients in NWChem: Estimating scalar relativistic effects using Douglas-Kroll contracted basis sets. J. Chem. Phys. 2001, 114, 48-53. 23. Feller, D. The role of databases in support of computational chemistry calculations. J. Comp. Chem. Phys. 1996, 17, 1571-1586. 24. Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, T. L. Basis set exchange: A community database for computational sciences. J. Chem. Inf. Model 2007, 47, 1045-1052. 25. Haeberlen, O. D.; Roesch, N. A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional methods: Application to AuH, AuCl, and Au2. Chem. Phys. Lett. 1992, 199, 491-496. 26. Nakajima, T.; Hirao, K. Numerical illustration of third-order Douglas-Kroll method: Atomic and molecular properties of superheavy element 112. Chem. Phys. Lett. 2000, 329, 511-516. 27. Nakajima, T.; Hirao, K. The higher-order Douglas-Kroll transformation. J. Chem. Phys. 2000, 113, 7786-7789. 28. Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158-6169. 29. Weber, R.; Yuwono, S.; Jeffrey, C.; Ouyang, J.; Schoendorff, G.; Wilson, A. K. Structure and energetics of LnX3 (Ln = La-Lu, X = F, Cl, Br) 70th Southwest Regional Meeting of the American Chemical Society (SWRM) 2014. 30. Molnár, J.; Hargittai, M. Prediction of the molecular shape of lanthanide trihalides. J. Phys. Chem. 1995, 99, 10780-10784. 31. Hargittai, M. Molecular structure of metal halides. Chem. Rev. 2000, 100, 2233-2301.

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32. Hastie, J. W.; Hauge, R. H.; Margrave, J. L. Geometries and entropies of metal trifluorides from infrared spectra: ScF3, YF3, LaF3, CeF3, NdF3, EuF3, and GdF3. J. Less-Common Met. 1975, 39, 309-334. 33. Slater, J. C.; Johnson, K. H. Self-consistent Xα cluster method for polyatomic molecules and solids. Phys. Rev. B 1972, 5, 844-853. 34. Vosko, S. H.; Wilk, L.; Nusair, M.; Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis. Can. J. Phys. 1980, 58, 12001211. 35. Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 1988, 38, 3098-3100. 36. Perdew, J. P. Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys. Rev. B 1986, 33, 8822-8824. 37. Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785-789. 38. Perdew, J. P. In Ziesche, P., Eschrig, H., Eds.; Electronic structure of solids ’91; Akademie Verlag: Berlin, 1991; 11. 39. Perdew, J. P; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865-3868. 40. Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids. Phys. Rev. Lett. 2003, 91, 146401. 41. Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Design of density functionals by combining the method of constraint satisfaction with parameterization for thermochemistry, thermochemical kinetics, and noncovalent interactions. J. Chem. Theory Comput. 2006, 2, 364-382. 42. Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648-5652. 43. Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J. Phys. Chem. 1994, 98, 11623-11627. 44. Becke, A. D. A new mixing of Hartree-Fock and local density-functional theories. J. Chem. Phys. 1993, 98, 1372-1377. 45. Becke, A. D. Density-functional thermochemistry. V. Systematic optimization of exchangecorrelation functionals. J. Chem. Phys. 1997, 107, 8554-8560. 46. Hamprecht, F. A.; Cohen, A. J.; Tozer, D. J.; Handy, N. C. Development and assessment of new exchange-correlation functionals. J. Chem. Phys. 1998, 109, 6264-6271. 47. Lynch, B. J.; Fast, P. L.; Harris, M.; Truhlar, D. G. Adiabatic connection for kinetics. J. Phys. Chem. A 2000, 104, 4811-4815. 48. Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes. J. Chem. Phys. 2003, 119, 12129-12137. 49. Zhao, Y.; Truhlar, D. G.; A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J. Chem. Phys. 2006, 125, 194101. 50. Swart, M.; Solà, M.; Bickelhaupt, F. M. A new all-round density functional based on spin states and SN2 barriers. J. Chem. Phys. 2009, 131, 094103.

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51. Swart, M.; Solà, M.; Bickelhaupt, F. M. Switching between OPTX and PBE exchange functionals. J. Comput. Methods Sci. Eng. 2009, 9, 69-77. 52. Yanai, T.; Tew, D. P.; Handy, N. C. A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51-57. 53. Peverati, R.; Truhlar, D. G. Improving the accuracy of hybrid meta-GGA density functionals by range separation. J. Phys. Chem. Lett. 2011, 2, 2810-2817. 54. Møller, C.; Plesset, M. S. Note on an approximation treatment for many-electron systems. Phys. Rev. 1934, 46, 618-622. 55. Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation. Chem. Phys. 2006, 124, 034108. 56. Konings, R. J. M. Beneš, O. Thermodynamic properties of the f-elements and their compounds. I. The lanthanide and actinide metals. J. Phys. Chem. Ref. Data 2010, 39, 043102. 57. James, A. M.; Lord, M. P. In Macmillan’s Chemical and Physical Data; Macmillan: London, 1992. 58. Ellis, H., Ed.; In Nuffield Advanced Science Book of Data; Longman: London, 1972. 59. Schall, H.; Dulick, M.; Field, R. W. The electronic structure of LaF: A multiconfiguration ligand field calculation. J. Chem. Phys. 1987, 87, 2989-2912. 60. Schamps, J.; Bencheikh, M.; Barthelat, J.-C.; Field, R. W. The electronic structure of LaO: Ligand field versus ab initio calculations. J. Chem. Phys. 1995, 103, 8004-8013. 61. Konings, R. J. M.; Beneš, O.; Kovács, A.; Manara, D.; Sedmidubský, D.; Gorokhov, L.; Iorish, V. S.; Yungman, V.; Shenyavskaya, E.; Osina, E. The thermodynamic properties of the felements and their compounds. Part 2. The lanthanide and actinide oxides. J. Phys. Chem. Ref. Data 2014, 43, 013101. 62. Ames, L. L.; Walsh, P. N.; White, D. Rare earths. IV. Dissociation energies of the gaseous monoxides of the rare earths. J. Phys. Chem. 1967, 71, 2707-2718. 63. Zmbov, K. F.; Margrave, J. L. In Mass spectrometric studies of scandium, yttrium, lanthanum, and rare-earth fluorides; Margrave, J. L., Ed.; Mass-spectrometry in inorganic chemistry; American Chemical Society: Washington, 1968; Vol. 72, 267-290. 64. Zmbov, K. F.; Margrave, J. L. Mass-spectrometric studies at high temperatures. XI. The sublimation pressure of NdF3 and the stabilities of gaseous NdF2 and NdF. J. Chem. Phys. 1966, 45, 3167-3170. 65. Zmbov, K. F.; Margrave, J. L. Mass spectrometric studies at high temperatures. XIII. Stabilities of samarium, europium and gadolinium mono- and difluorides. J. Inorg. Nucl. Chem. 1967, 29, 59-63. 66. Yungman, V. S. In Thermal constants of substances; Wiley: New York, 1999, Vol. 6. 67. Becke, A. D. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys. 2014, 140, 18A301.

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Tables and Figures Figure 1. Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the Ln54HFa subset containing lanthanides formally in the common 3+ oxidation state. MSDs and MADs are in kcal mol-1.

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Figure 2. Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the Ln54HFb subset containing lanthanides formally in 1+ and 2+ oxidation states. MSDs and MADs are in kcal mol-1.

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Figure 3. Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the combined Ln54HF subset (Ln54HFa and Ln54HFb). MSDs and MADs are in kcal mol-1.

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Figure 4. Performance of density functionals for the computation of bond dissociation energies, D0, for molecules in the Ln54D0 subset of diatomic molecules. MSDs and MADs are in kcal mol-1.

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Figure 5. Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), and bond dissociation energies, D0, for molecules in the full Ln54 set of molecules. MSDs and MADs are in kcal mol-1.

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Table 1. Performance of the M06 family of density functionals for the Ln54HFa, Ln54HFb, and Ln54D0 subsets of molecules and for the full Ln54 molecule set. An increase in the amount exact exchange in the functional results in an increase in the MAD. MADs are in kcal mol-1. Mean Absolute Deviation (kcal mol-1) % Exact Exchange Ln54HFa Ln54HFb Ln54D0 Ln54 M06-L 0 29.5 18.2 24.2 22.7 M06 27 33.1 22.0 33.2 28.3 M06-2X 54 46.5 31.8 46.0 39.9

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Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the Ln54HFa subset containing lanthanides formally in the common 3+ oxidation state. MSDs and MADs are in kcal mol-1. 352x264mm (72 x 72 DPI)

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Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the Ln54HFb subset containing lanthanides formally in 1+ and 2+ oxidation states. MSDs and MADs are in kcal mol-1. 352x264mm (72 x 72 DPI)

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Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), for molecules in the combined Ln54HF subset (Ln54HFa and Ln54HFb). MSDs and MADs are in kcal mol-1. 352x264mm (72 x 72 DPI)

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Performance of density functionals for the computation of bond dissociation energies, D0, for molecules in the Ln54D0 subset of diatomic molecules. MSDs and MADs are in kcal mol-1. 352x264mm (72 x 72 DPI)

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Performance of density functionals for the computation of enthalpies of formation, ∆Hf(298), and bond dissociation energies, D0, for molecules in the full Ln54 set of molecules. MSDs and MADs are in kcal mol-1. 352x264mm (72 x 72 DPI)

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